An object with a mass of #4 kg# is on a plane with an incline of # - pi/4 #. If it takes #7 N# to start pushing the object down the plane and #6 N# to keep pushing it, what are the coefficients of static and kinetic friction?

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Al E. Share
Feb 22, 2018

Consider your situation,

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where:

#theta = pi/4 = 45°#

Moreover, recall,

#SigmavecF = ma#

Let's determine the normal force,

#SigmaF_y = F_N - mgcostheta = 0#
#=> F_N = mgcostheta#

and recall that to overcome the static friction a force equal to that friction must be exerted on the object.

Hence,

#SigmaF_x = F_P + mgsintheta = mu_s * F_N#

#=> mu_s = (F_P +mgsintheta)/(mgcostheta) approx 1.25#

If we wish to keep pushing the object down the incline, we need to push it with a force greater than or equal to its kinetic friction. Assuming that we are pushing the object with the least amount of force before friction overcomes us, then,

#SigmaF_x = F_P + mgsintheta = mu_k * F_N#

#=> mu_k = (F_P + mgsintheta)/(mgcostheta) approx 1.22#

Note I assumed gravity aided us in each case due to the incline. The negative angle of inclination is kind of a curve ball that gets students who don't know the material that well, it kind of looks like this on a coordinate plane,

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Thus, I took it to mean in that direction, rather than the one depicted above.

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